Realistic Parameters Adoption to Solve Rock Engineering Problems

Abstract

Strength and modulus of rock mass as obtained from RMR, Q and GSI have been examined with reference to modulus ratio, M_rj, for their reliability. The design parameters adopted in some case studies based on these rock mass classifications are presented. The modulus ratios in these case studies are found to be much higher than those of the corresponding values of intact rocks, even after back analyses. Based on joint factor, J_f, compressive strength, modulus, cohesion and friction angle were estimated and applied in the analyses of a few cases. The predictions of deformations agreed well with the field measurements. Based on extensive experimental data of jointed specimens of rock and rock-like materials, a joint factor, J_f, was defined as a weakness coefficient in rock mass compared to the corresponding intact rock. J_f is linked to the strength, modulus and modulus ratio of rock mass. The modulus ratio, M_rj, of rocks is less than the modulus ratio of intact rock. The M_rj concept has been adopted to present a unified classification for intact rocks and rocks masses, to define soil that rock boundary and penetration rate of TBMs.

Introduction

Soil and rock are geotechnical materials formed through very complex processes. Soil is treated most often as a continuum, i.e., homogeneous and isotropic, for the purpose of analysis and design, even though it is a particulate material. An undisturbed specimen can be obtained from the field, or a remolded specimen is prepared in the laboratory to the required density and water content and tested to assess the required soil parameters under relevant drainage conditions, confining pressures either in axisymmetrical or in asymmetrical stress states. A soil test specimen is so chosen that its minimum dimension is many times larger than the maximum soil particle size to get representative and consistent results.

It is not so in the case of rock masses. They are discontinuous, non-homogeneous, anisotropic and prestressed. Collection of undisturbed specimen of rock mass to test in a laboratory is considered uneconomical and mostly not practicable. In some cases, large field shear and plate load tests are conducted to assess rock mass properties. Even these tests are being considered time-consuming and expensive. So tests are conducted as a routine on intact rock specimens in a laboratory to arrive at the upper bound values of rock parameters. Attempts have been made to correlate strength and modulus of intact rock with those of rock mass through rock mass classifications. These correlations are often being adopted to assess compressive strength (σ_cj), modulus (E_j), cohesion (c_j), friction angle ($\phi_{\text{j}}$) and to predict stress strain response of rock mass during the last four decades; subscript j refers to rock mass.

Hypothetical stress–strain curves for three different rocks are presented in Fig. 1. Curves OA, OB and OC represent three stress–strain curves with failure occurring at A, B and C, respectively. Curves OA and OB have same modulus but different strengths and strains at failure, whereas the curves OA and OC have same strength but different moduli and failure strains. So neither strength nor modulus alone could be chosen to represent the overall quality of the rock. Therefore, strength and modulus together will give a realistic understanding of the rock response for engineering usage. This approach to define the quality of intact rocks was proposed by Deere and Miller [1] by considering the modulus ratio, $M_{\text{ri}} = E_{\text{i}} /\sigma_{\text{ci}}$, where E_i = tangent modulus at 50% of failure strength, σ_ci = compressive strength at failure, and i is subscript for intact rock.

Fig. 1

Hypothetical stress–strain curves

Modulus Ratio Concept

Deere and Miller presented a classification of intact rocks based on modulus value (E_i) at 50% of the failure stress and the unconfined compressive strength (σ_ci). Vast experimental data of 613 rock specimens from different locations covering 176 igneous, 193 sedimentary, 167 metamorphic and 77 limestones and dolomites were presented by them to classify intact rocks on the basis of (σ_ci) and modulus ratio, $M_{\text{ri}} ( = E_{\text{i}} /\sigma_{\text{ci}} )$. It is found that for basalts and limestones, one could expect M_ri values up to 1600, whereas for shales this value could be close to 60. Even weathered Keuper showed M_ri close to 50. Only chalks with high porosity have shown M_ri greater than 1000 [2] due to high modulus and low strength.

Rock Mass Parameters from Classifications

The most commonly adopted rock mass classifications are RMR, Q and GSI systems at present. Significant contributions have come from Bieniawski [3] and Barton et al. [4] based on their vast experience in a number of tunnels. Erlier GSI classification by Hoek [5] and Hoek and Brown [6] was a combination of RMR and Q approaches with some modification. The value of GSI was same as RMR as suggested by Bieniawski [7] for RMR > 18 and, for GSI < 18, Q values were considered for the rock mass. Later on, they suggested to arrive at GSI based on fracture intensity and degree of joint roughness matching with sketches and table. Often designers adopt one or more classifications to arrive at conservative designs.

Strength and Modulus from RMR

Bieniawski [3] suggested shear strength parameters, c_j (cohesion) and ϕ_j (friction angle), where subscript j refers to rock mass, for five levels of rock mass classes (Table 1). These parameters have been in use for over 4 decades. With these values of c_j and ϕ_j, the uniaxial compressive strengths (σ_cj) of the mass are calculated as per Mohr–Coulomb criterion, Eq. (1) and are referred in Table 2

Table 1 RMR strength values for rock masses [3]

Table 2 M_rj values with RMR for rock masses

$$ \sigma_{\text{cj}} = {{2c_{\text{j}} \cos \phi_{\text{j}} } \mathord{\left/ {\vphantom {{2c_{\text{j}} \cos \phi_{\text{j}} } {(1 - \sin \phi_{\text{j}} )}}} \right. \kern-0pt} {(1 - \sin \phi_{\text{j}} )}}. $$

(1)

Table 2 includes the estimated values of E_j from Eq. (2) as per Serafim and Pereira [8],

$$ E_{\text{j}} = 10^{{({\text{RMR}} - 10)/40}} ,\,{\text{GPa}}. $$

(2)

It is obvious from this table that the modulus ratios, $M_{\text{rj}} = E_{\text{j}} /\sigma_{\text{cj}}$, decrease with the decrease in RMR, but the values are extremely high for rock masses, as compared to intact rocks [1].

Strength and Modulus from Q

Barton [9] suggested modification to the earlier Q values [4] by considering the influence of uniaxial compressive strength of the intact rock (σ_ci) in the following form

$$ Q_{\text{c}} = \, Q\sigma_{\text{ci}} / \, 100 $$

(3)

and recommended Q_c values for estimating the compressive strength and modulus of rock mass as

$$ \sigma_{\text{cj}} = 5\gamma Q_{\text{c}}^{l/3} ,\,{\text{MPa}}\;{\text{and}} $$

(4)

$$ E_{\text{j}} = 10 \, Q_{\text{c}}^{l/3} ,\,{\text{GPa}}, $$

(5)

where γ density of rock mass in g/cc or t/m³. Equation (4) suggests that Q is linked to the compressive strength of rock mass through the intact rock strength. On the contrary, the modulus of rock mass, E_j, is not linked to the modulus of the intact rock through Q.

Another important recommendation of Barton [9] is to assess c_j and ϕ_j of rock mass from the following expressions,

$$ c_{\text{j}} = \left( {{\text{RQD}}/J_{\text{s}} } \right)\left( {1/{\text{SRF}}} \right) \, (\sigma_{\text{ci}} /100),\,{\text{MPa}} $$

(6)

$$ \phi_{\text{j}}^{ \circ } = \tan^{ - 1} (J_{\text{r}} J_{\text{w}} /J_{\text{a}} ), $$

(7)

where J_s is joint set number (Barton uses J_n), SRF is the stress reduction number, J_r is joint roughness number, J_w is for seepage and its pressure, and J_a is joint alteration number.

With the data provided in Tables 6 and 7 of Barton [9], the values of compressive strength of rock mass are calculated as per Mohr–Coulomb criterion, Eq. (1) with c_j and ϕ_j, and are referred to as σ_cj2 in Table 3 of this paper. The data in this table suggest that σ_cj2 values differ significantly from the suggested values (σ_cj1) by Eq. (4). The ratio of σ_cj1/σ_cj2 varies from 1:7 to 54:1 depending upon the value of Q_c. This table also gives the values of E_j as per Barton [9]. The values of modulus ratio, M_rj, are more or less constant and are around 800; in fact, Eqs. (4, 5) give this ratio as 800 for a value of density, γ = 2.5 g/cc, irrespective of Q_c varying from 0.008 to 100, i.e., whether the rock is intact, jointed, isotropic or anisotropic.

Table 3 Values of σ_cj, c_j, ϕ_j and E_j as per Barton [9] for σ_ci = 100 MPa, γ = 2.5 g/cc

Strength and Modulus from GSI

Hoek [5] and Hoek and Brown [6] advocate the adoption of Geological Strength Index, GSI, to estimate the material parameters, m_j and s_j, of the Hoek–Brown failure criterion to predict strength under any desired confining pressure. GSI was based on both RMR and Q systems with some modifications mainly to estimate the compressive strength of rock mass. Their original expression is Eq. (8).

$$ {{\sigma_{\text{cj}} } \mathord{\left/ {\vphantom {{\sigma_{\text{cj}} } {\sigma_{\text{ci}} }}} \right. \kern-0pt} {\sigma_{\text{ci}} }} = \sqrt {s_{\text{j}} } ;\quad s_{\text{j}} = \exp \left[ {\left( {{\text{GSI}} - \, 100} \right)/9} \right]. $$

(8)

There have been modifications subsequently to estimate s_j and √s_j by considering disturbance factor, D. For estimating the deformation modulus, Hoek [5] recommends the use of Eq. (2) using RMR as per Bieniawski [3] and not the GSI value.

The values of GSI, s_j and E_j given in Table 5 of Hoek [5] have been considered in calculating the values of modulus ratio, M_rj, in Table 4 of this paper. The values of M_rj are surprisingly high, ranging from 1500 to 1720 with an average value of 1621 for GSI varying from 85 to 34. These values of M_rj do not decrease, with the decrease in GSI value.

Table 4 Prediction of σ_cj, E_j and M_rj for σ_ci = 100 MPa.

Parameters Used in Case Studies

The design parameters, i.e., compressive strength and modulus of rock mass (E_j), adopted in some of the recent projects based on rock mass classifications have been checked with M_r concept. Only those cases, in which the parameters of intact and rock mass are available, have shown that M_rj values are much higher than M_ri. This is contrary to the experimental evidence of jointed specimens. In the following, these cases are briefly presented.

1. In the underground pump storage development of Rio Grande No. 1, Argentina, massive gneiss Pelado was encountered [10]. The RQD of the rock mass varied from 65 to 90%. The modulus of intact rock, E_i varied from 30 to 60 GPa, and compression test gave σ_ci = 140 MPa. But the triaxial compression test gave σ_ci = 110 MPa with c_i = 20 MPa and $\phi_{\text{j}} = 50^{ \circ }$. The M_ri would range from 273 to 546. The field shear test gave c_j = 0.34 MPa and $\phi_{\text{j}} = 30^{ \circ }$ (peak) resulting in $\sigma_{\text{cj}} = 1.25\, {\text{MPa}}$ as per Mohr–Coulomb theory. The modulus of deformation, $E_{\text{d}}$, from plate loading test varied from 40 to 90 GPa from loading and unloading cycles; average being $E_{\text{d}} = 60 \,{\text{GPa}}$, adopted in the analysis. Finite element (FE) and boundary element (BE) analyses were conducted with K₀ varying from 0.5 to 2.0, considering the rock as elastic medium. The $M_{\text{rj}} = 4800$ and $M_{\text{rj}} / M_{\text{ri}} = 8.79$ by considering the maximum $M_{\text{ri}}$. These values are rather very high; the $M_{\text{rj}} / M_{\text{ri}}$ should have been less than 1.0. They have indicated that the measured deformations were only twice the calculated values.

2. The rock encountered in the Masua mine, Italy, was dolomite limestone with ${\text{RMR}} = 80,\; \sigma_{\text{ci}} = 87.8 \,{\text{MPa}},\; E_{\text{i}} = 78\,{\text{GPa}},\; \sigma_{\text{t}} = 5.6\,{\text{MPa}},\; c_{\text{i}} = 31\,{\text{MPa}},\; \phi_{\text{i}} = 45^{ \circ }$ and $M_{\text{ri}} = 888,$ [11]. From the field plate loading test for the predominantly isotropic rock, $E_{\text{d}}$ varied from $37.5 \pm 5.5\, {\text{GPa}}.$. The final RMR chosen was $68 \pm 10.8$, and from GSI, $\sigma_{\text{cj}}$ estimated was $15.8123 \pm 0.0046\, {\text{MPa}}$. The $E_{\text{j}}$ varied between 32 and 43 GPa with average being 37.5 GPa. The ratio $M_{\text{rj}} / M_{\text{ri}}$ is, therefore, for maximum value of $M_{\text{rj}}$, 3.07 and for minimum is 2.28. By adopting discontinuity method for 3D analysis and FEM for 2D analysis, it has been mentioned that the deformations on the hanging wall of the southern open slope were of the same range of the predicted values.

3. Hoek and Moy [12] dealt with various aspects of “Power house caverns in weak rock.” For siltstone, $\sigma_{\text{ci}} = 100\, {\text{MPa}},\; {\text{RMR}} = 48,\; E_{\text{d}} = 8.9\, {\text{GPa}},\; s_{\text{j}} = 0.003 \;{\text{resulting}}\; \sigma_{\text{cj}} = 5.477\, {\text{MPa}};\;{\text{No}}\;E_{\text{i}}$ value is available. $M_{\text{rj}} = 1625,$ appears to be rather high for RMR = 48. Even by assuming a high value of $M_{\text{ri}} = 500$, $M_{\text{rj}} / M_{\text{ri}} = 3.25.$

4. In their contribution on “Practical estimates of rock mass strength,” Hoek and Brown [6] recommendations indicate

• (a) For good quality rock (GSI = 75), $\sigma_{\text{cj}} = 64.8 \,{\text{MPa}}, \;E_{\text{j }} = 42 \,{\text{GPa}}$, M_rj = 648

• (b) For Avrg. quality rock (GSI = 50), $\sigma_{\text{cj}} = 13.0\, {\text{MPa}},\;E_{\text{j }} = 9.0\,{\text{GPa}}$, M_rj = 692

• (c) For poor quality rock (GSI = 30), $\sigma_{\text{cj}} = 1.7 \,{\text{MPa}},\;E_{\text{j }} = 1.4 \,{\text{GPa}}$, M_rj = 824

• (d) For Braden braccia, El Teniete mine, Chile

  GSI = 75, $c_{\text{j}} = 4.32 \,{\text{MPa}},\;\phi_{\text{j }} = 42^{ \circ } ,\; \sigma_{{{\text{cj}} }} = 19.4 \,{\text{MPa}}, \;E_{\text{j}} = 30\,{\text{GPa}}$, M_rj = 1546

• (e) For Nathpa Jhakri HE Project, India,

  Quartz mica schist (GSI = 65), $c_{\text{j}} = 2.0 \,{\text{MPa}},\;\phi_{\text{j }} = 40^{ \circ } , \;\sigma_{\text{cj }} = 8.2 \,{\text{MPa}},\; E_{\text{j}} = 13 \,{\text{GPa}}$, M_rj = 1585

• (f) Athens schist—decomposed, GSI = 20

  $c_{\text{j}} = 0.09 - 0.018 \,{\text{MPa}},\;\phi_{\text{j}} = 24^{ \circ } , \;\sigma_{\text{cj }} = 0.27 - 0.53\,{\text{MPa}}, \;E_{\text{j}} = 398 - 562\,{\text{MPa}}$

  M_rj = 1060 (min) and 1474 (max.)

• (g) Yacambu Quibor tunnel, Venezuela,

  For poor quality graphitic phyllite, GSI = 24,

  $c_{\text{j}} = 0.34\, {\text{MPa}},\;\phi_{\text{j }} = 24^{ \circ } , \;\sigma_{\text{cj }} = 1.0\,{\text{MPa}}, \;E_{\text{j}} = 870\,{\text{MPa}}$.

Note: M_rj values are rather high; stronger rocks having lower and weaker ones having higher values.

5. For the Mingtan Pump Storage Project, Taiwan, underground power cavern, 22 m wide, 46 m high, 158 m long, and a transformer hall, 13 m wide, 20 m high, 172 m long, are located at a depth of 300 m below the ground level in jointed sandstone and bedded sandstone [13].

• 1. For jointed sandstone, $\sigma_{\text{ci}} = 166 \,{\text{MPa}}, \;E_{\text{i}} = 22.3\, {\text{GPa}},\;M_{\text{ri}}$ = 134

• 2. For bedded sandstone, $\sigma_{\text{ci}} = 66\,{\text{MPa}}, \;E_{\text{i}} = 12.8\, {\text{GPa}},\;M_{\text{ri}}$ = 194.

These are average values for rock cores.

The average RMR values for jointed sandstone and for bedded sandstone have been 69 and 58, respectively. They have adopted for:

• 1. Jointed sandstone: $\sigma_{\text{cj}} = 45 - 65\,{\text{MPa,}}\;E_{\text{j}} = 29.85\, {\text{GPa}}, \;{\text{so}}\; M_{\text{rj }}$ = 543 and $M_{\text{rj}} / M_{\text{ri}} = 4.0$

• 2. Bedded sandstone: $\sigma_{\text{cj}} = 11.48 \,{\text{MPa}}, \;E_{\text{j}} = 15.85 \;{\text{GPa}}, \;{\text{so}}\; M_{\text{rj}}$ = 1381 and $M_{\text{rj}} / M_{\text{ri}} = 7.1$.

When plate loading tests were conducted, the modulus of deformations ($E_{\text{d}}$) was much lower than the modulus as per Serafim and Pereira. For the jointed sandstone, $E_{\text{d}}$ = 4.15 GPa and for bedded sandstone $E_{\text{d}}$ = 2.9 GPa.

By adopting 2D finite difference programme FLAC, the progressive response of rocks was treated as elasto-plastic materials as per Mohr–Coulomb theory. Iterated analysis was carried out by lowering the parameters of rocks both intact and mass, to obtain deformations to match the measured ones. Finally, computed back analysis data were:

• 1. Jointed sandstone:

  $\sigma_{\text{ci}} = 100\,{\text{MPa}},\; m_{\text{j}} = 4.298,\;s_{\text{j }} = 0.02047$ giving $\sigma_{\text{cj}} = 14.3\,{\text{MPa}}\;{\text{and}}\;E_{j } = 4.5 \,{\text{GPa}},\;{\text{so}}\; M_{\text{rj}} = 315,$ $M_{\text{rj}} / M_{\text{ri}} = 2.4$.

• 2. For the bedded sandstone,

  $\sigma_{\text{ci}} = 100 \,{\text{MPa}}, \;m_{\text{j}} = 1.519,\;s_{\text{j }} = 0.00211\;{\text{giving}}\; \sigma_{\text{cj}} = 4.59\, {\text{MPa}}\;{\text{and}}\;E_{\text{j }} = 2.5\, {\text{GPa}}$,${\text{so}} \;M_{\text{rj}} = 545,$ $M_{\text{rj}} / M_{\text{ri}} = 2.8$.

In both the cases of sandstones, the $M_{\text{rj}}$ values are higher than $M_{\text{ri}}$ values even after conducting iterative analysis to match the measured deformations.

6. To provide appropriate support in the chromite mine, Ermekov et al. [14] carried out physical model studies by adopting equivalent material modeling with a scale of 1:50. For the horizontal working of chromite in the western Kazakhstan, they adopted three support systems: (i) framed arch, (ii) anchor supports with framed arch and (iii) injected anchored support with framed arch. The stress state at the depth of 135 m in the mine was estimated as σ₁ = 16 MPa, σ₂ = 14 MPa and σ₃ = 11 MPa.

The modulus of chromite cores, $E_{\text{i }} = 620\, {\text{GPa}}, \;\sigma_{\text{ci}} = 31\, {\text{MPa}},\;M_{\text{ri}} = 2000.$ For the rock cores surrounding the chromite, $\sigma_{\text{ci}} = 66\,{\text{MPa}},\;E_{\text{i}} = 46\, {\text{GPa}}, \;M_{\text{ri}} = 697.$ The $\sigma_{\text{cj}}$ of chromite = 10 MPa, $E_{\text{j }} = 2.0 \,{\text{GPa}}, \;M_{\text{rj}} = 200$. For rock mass, $\sigma_{\text{cj}} = 20\,{\text{MPa}},$ $E_{\text{j }} = 1.5\, {\text{GPa}}, \;M_{\text{rj}} = 75.$ Therefore, for chromite $M_{\text{rj}} / M_{\text{ri}} = 0.1$ and for the rock mass, $M_{\text{rj}} / M_{\text{ri}} = 0.11.$ The properties chosen for chromite and rock mass not based on RMR, Q or GSI seem to be in order, i.e., $M_{\text{rj}} / M_{\text{ri}}$ values are less than 1.0.

7. For the large span underground storage project, a cavern 25 m wide, 12 m high and 100 m long is located in Bukit Timah granite. Rock reinforcement was adopted as per Q-system and numerical modeling was also carried out [15]. Measurements and numerical analysis showed high horizontal stresses favouring the stability of the cavern. The crown showed upward deformation of the order of 1.0 mm only. σ_ci = 108–225 MPa (average 164 MPa, E_i = 49.3–113.3 GPa) (average 65.6 GPa). The estimated $\sigma_{\text{cj}}$ = 45–65 MPa (average 55 MPa) and E_j = 40–50 GPa) (average 45 GPa). The $M_{\text{ri}} = 400, \; M_{\text{rj}} = 818$ and $M_{\text{rj}} / M_{\text{ri}} = 2.04,$ which is more than 1.0.

8. For 1000 MW Masjed-e-Soloiman HEPP, Iran, Stabel and Samani [16] carried out 2D elasto-plastic hybrid FEM and BEM analyses on the powerhouse cavern, 30 m span, 50 m high and 151 m long located in mudstone consisting of layers of conglomerate, sandstone, siltstone and claystone. The rock core properties from laboratory tests of 1991 were:

• 1. Conglomerate: $\sigma_{\text{ci}} = 57 \,{\text{MPa}},\;E_{\text{i }} = 45\,{\text{GPa}}, \;{\text{so}}\;M_{\text{ri}} = 789$

• 2. Sandstone: $\sigma_{\text{ci}} = 67 \,{\text{MPa}},\;E_{\text{i }} = 25\,{\text{GPa}}, \;{\text{so}}\;M_{\text{ri}} = 373$

• 3. Siltstone: $\sigma_{\text{ci}} = 39 \,{\text{MPa}},\;E_{\text{i }} = 13\,{\text{GPa}}, \;{\text{so}}\;M_{\text{ri}} = 333$

• 4. Claystone: $\sigma_{\text{ci}} = 23 \,{\text{MPa}},\;E_{\text{i }} = 8\,{\text{GPa}}, \;{\text{so}}\;M_{\text{ri}} = 348$.

In the analyses, the following properties of rock layers were chosen:

• 1. Conglomerate: $\phi_{\text{j}} = 43^{ \circ } ,\; c_{\text{j}} = 2.87\,{\text{MPa,}}\; \sigma_{\text{cj}} = 13.2\, {\text{MPa}},\;E_{\text{j}} = 15\,{\text{GPa}}$, so, ${\text{M}}_{\text{rj}} = 1136$

• 2. Sandstone $\phi_{\text{j}} = 38^{ \circ } ,\; c_{\text{j}} = 1.67\,{\text{MPa,}}\; \sigma_{\text{cj}} = 6.85\, {\text{MPa}},\;E_{\text{j}} = 7.0\,{\text{GPa}}$, so,${\text{M}}_{\text{rj}} = 1022$

• 3. Siltstone: $\phi_{\text{j}} = 30^{ \circ } ,\; c_{\text{j}} = 0.73\,{\text{MPa,}}\; \sigma_{\text{cj}} = 2.53\, {\text{MPa}},\;E_{\text{j}} = 6.0\,{\text{GPa}}$, so, ${\text{M}}_{\text{rj}} = 2372$

• 4. Claystone: $\phi_{\text{j}} = 24^{ \circ } ,\; c_{\text{j}} = 0.50\,{\text{MPa,}}\; \sigma_{\text{cj}} = 1.54\, {\text{MPa}},\;E_{\text{j}} = 6.0\,{\text{GPa}}$, so, $M_{\text{rj}} = 3896$

The $\sigma_{\text{cj}}$ values are calculated from $c_{\text{j}}$ and $\phi_{\text{j}}$ as per Mohr–Coulomb criterion. Therefore, for:

• 1. Conglomerate: $M_{\text{rj}} / M_{\text{ri}} = 1.44$

• 2. Sandstone: $M_{\text{rj}} / M_{\text{ri}} = 2.74$

• 3. Siltstone: $M_{\text{rj}} / M_{\text{ri}} = 7.12$

• 4. Claystone: $M_{\text{rj}} / M_{\text{ri}} = 11.20$.

These values are rather high.

9. Intake tunnel, Karuna III, HEPP, Iran: the analysis of the tunnel was carried out using 2D, UDEC and 3D elastic programmes [17]. Measured deformations agreed with the estimated values from back analysis. Back analysis was carried out using direct method UDEC software, and σ_cj and E_j were estimated. For Marly limestone/marl,

$$ c_{\text{j}} = 0.6 \,{\text{MPa}}, \phi_{\text{j}} = 30^{ \circ } ,\; \sigma_{\text{cj}} = 2.08 \,{\text{MPa}},\;E_{\text{j }} = 6.0\, {\text{GPa}} . $$

By considering the average values of the rocks,

σ_ci = 70 MPa, E_i = 10.25 GPa, M_ri = 146 and M_rj = 2855, so, M_rj/M_ri = 19.8.

10. For the underground pumped powerhouse caverns, namely Samragjin, Muju, Sanchung, Yangyang and Chungsong, constructed during 1973–2003, the numerical softwares used were (1) visco-elastic FEM, (2) hybrid combining FEM and BEM and (3) elasto-plastic FEM [18]. The rock mass parameters were assumed by applying reduction factors to modulus (0.16–0.55) to cohesion (0.12–0.22), and to friction angle (0.6–0.87) to the corresponding values of intact rock cores. Mainly Mohr–Coulomb failure criterion was adopted. Detailed calculations with the parameters adopted for the analyses gave the following:

Rock type	Cavern	$M_{\text{rj}} /M_{\text{ri}}$	Convergence, mm
Granitic Gneiss	Muju	1.60	19.0
Porphyroblastic Gneiss	Sanchung	2.87	54.4
Porphyroblastic Gneiss, Granite Gneiss	Yangyang	3.83	55.0
Sandstone, Arkosic sandstone	Chungsong	3.30	50.0

1. For Samragjin cavern, reduction factors were not included in their paper

11. A synthetic rock mass (SRM) approach, which can take care of anisotropy and scale effects, was adopted by Carvalho et al. [19] to predict unconfined compressive and triaxial strengths. Clark [20], quoted by Lorig [21], using FLAC (ITASCA 2005) constructed SRM model based on actual scaled distribution of joints, and predicted strength in unconfined state with RMR covering anisotropy and scale effects as shown in Table 5. J_f and strength ratios ($\sigma_{\text{cj}} /\sigma_{\text{ci}}$) are also indicated. The SRM values agree reasonably well with the experimental findings based on joint factor, J_f.

Table 5 Results of σ_cj/σ_ci from SRM and J_f [20]

12. Read [22], from synthetic rock mass (SRM) model for carbonatite 2D joints, gave

$$ \sigma_{\text{ci}} = 140 \;{\text{MPa}},\;E_{\text{i }} = 60\, {\text{GPa}}, \;{\text{resulting}}\; M_{\text{ri}} = 430 $$

• 1.
  $$ {\text{For}} \;20\,{\text{m}}\;{\text{length }}\;{\text{of}}\;{\text{joint}},\; \sigma_{\text{cj}} = 80\, {\text{MPa}},\; E_{\text{j }} = 30\, {\text{GPa}}, \;M_{\text{rj}} = 375 $$
• 2.
  $$ {\text{For}} \;40\,{\text{m}}\;{\text{length }}\;{\text{of}}\;{\text{joint}},\; \sigma_{\text{cj}} = 60\, {\text{MPa}},\; E_{\text{j }} = 25\, {\text{GPa}}, \;M_{\text{rj}} = 416 $$
• 3.
  $$ {\text{For}} \;80\,{\text{m}}\;{\text{length }}\;{\text{of}}\;{\text{joint}},\; \sigma_{\text{cj}} = 70\, {\text{MPa}},\; E_{\text{j }} = 30\, {\text{GPa}}, \;M_{\text{rj}} = 429 $$

Here, the $M_{\text{rj}}$ values from 2D joint simulated rocks gave slightly lower values compared to the intact material. The SRM model seems to predict compressive strength and modulus values of simulated rock mass reasonably well in 2D case indicating negligible scale effect for the lengths of joints considered.

13. A stochastic analysis was carried out to estimate σ_cj and E_j of three grades of Ankara andesites by calculating the influence of correlations between relevant distributions on the simulated RMR values [23]. The model was also used in Monte Carlo simulation to estimate possible ranges of the Hoek–Brown strength parameters.

• From minimum strength and modulus,

• $$ {\text{Grade}}\;{\text{A}}:M_{\text{ri}} = 520,\;M_{\text{rj}} = 7000;\;M_{\text{rj}} /M_{\text{ri}} = 13.46 $$
• $$ {\text{Grade}}\;{\text{B}}:M_{\text{ri}} = 470,\;M_{\text{rj}} = 2113;\;M_{\text{rj}} /M_{\text{ri}} = 4.50 $$
• $$ {\text{Grade}}\;{\text{C}}:M_{\text{ri}} = 359,\;M_{\text{rj}} = 1311;\;M_{\text{rj}} /M_{\text{ri}} = 3.65 $$

• From maximum strength and modulus,

• $$ {\text{Grade}}\;{\text{A}}:M_{\text{ri}} = 425,\;M_{\text{rj}} = 1568;\;M_{\text{rj}} /M_{\text{ri}} = 3.69 $$
• $$ {\text{Grade}}\;{\text{B}}:M_{\text{ri}} = 431,\;M_{\text{rj}} = 1378;\;M_{\text{rj}} /M_{\text{ri}} = 3.20 $$
• $$ {\text{Grade}}\;{\text{C}}:M_{\text{ri}} = 326,\;M_{\text{rj}} = 963;\;M_{\text{rj}} /M_{\text{ri}} = 2.95 $$

14. Waitaki dam block No. 10, New Zealand: A 3D FEM analysis was carried out for the Waitaki dam Richards and Read [24]. The tiltmeter deformations under the block No. 10 were matched to obtain in situ modulus E_j. The ratio E_j/E_i was 0.15 for GSI = 20 of class II Greywacke. This value (0.15) has been found to be very high, about 5 times, even for disturbance factor, D = 0 as per Hoek and Diederichs [25]. The intact rock properties were

$\sigma_{\text{ci }} = 50 - 60 \,{\text{MPa}},\;E_{\text{i }} = 70 \,{\text{GPa}};$ M_rj = 1167 by considering σ_ci = 60 MPa.

As per Hoek and Brown [6] for GSI = 20,

σ_Cj = 0.704 MPa by considering σ_Ci = 60 MPa, E_j = 10 GPa, M_rj = 14,205, therefore M_rj/M_ri = 12.2.

15. Power house cavern, Rogun project, Kajikistan:

To predict deformations of roof and side walls using 3D FEM with M-C criterion, Bronshteyn et al. [26] adopted reduced c_i, ϕ_i and E_i as indicated here

$$ \begin{aligned} {\text{For}}\;{\text{sandstone}}:\;c & = 3.0\;{\text{to}}\;1.1\,{\text{MPa}}\left( {{\text{F}} . {\text{S}}.\, = \,2.7} \right), \\ \phi^\circ & = \,50\;{\text{to}}\;38\left( {{\text{F}} . {\text{S}} .\;{\text{to}}\;\tan \,\phi : \, 1.53} \right) \\ E & = 9000\;{\text{to}}\;5000\,{\text{MPa}}\,({\text{F}} . {\text{S}} .= 1.4) \\ M_{\text{ri}} & = 546,\;M_{\text{rj}} = 1109;\;M_{\text{rj}} /M_{\text{ri}} = 2.03. \\ \end{aligned} $$

$$ \begin{aligned} {\text{For}}\;{\text{aleurolites}}:\;c & = 2\;{\text{to}}\;0.5\,{\text{MPa }}\left( {{\text{F}} . {\text{S}} .\, = \,4.0} \right) \\ \phi^\circ & = 45\;{\text{to}}\;32.5 \, \left( {{\text{F}} . {\text{S}}.\;{\text{to}}\;\tan \phi :1.56} \right) \\ E & = 5500\;{\text{to}}\;2670\,({\text{F}} . {\text{S}} .= 2.06) \\ M_{\text{ri}} & = 569,\;M_{\text{rj}} = 1467;\;M_{\text{rj}} /M_{\text{ri}} = 2.58. \\ \end{aligned} $$

From a few case studies presented in the foregoing, it is obvious that strength and modulus adopted conclusively indicate M_rj/M_ri greater than 1.0.

Strength and Modulus from Join Factor J _f

Based on the extensive experimental results in uniaxial compression on jointed rocks and rock-like materials, the compressive strength of jointed mass is suggested close to the minimum values by Eq. (9) and the corresponding modulus by Eq. (10) [27],

$$ \sigma_{\text{cj}} /\sigma_{\text{ci}} = \exp \left[ { - \, 0.008\,J_{\text{f}} } \right]\;{\text{and}} $$

(9)

$$ E_{\text{j}} /E_{\text{i}} = \exp \left[ { - 0.0115\,J_{\text{f}} } \right], $$

(10)

wherein J_f is a joint factor defined by Eq. (11)

$$ J_{\text{f}} = \, J_{n} /n.r, $$

(11)

where J_n joint frequency, i.e, number of joints/meter, which takes care of RQD and joint sets and joint spacing; n inclination parameter depends on the inclination of sliding plane with respect to the major principal stress direction; the joint or set which is closer to (45 − ϕ_j/2)° with the major principal stress will be the most critical one to experience sliding at first; r a parameter for joint strength; it takes care of the influence of closed or filled-up joint, thickness of gouge, roughness, extent of weathering of joint walls and cementation along the joint. This factor could be assessed in terms of an equivalent value of friction angle along the joint as tan ϕ_j = τ_j/σ_nj obtained from shear tests, in which τ_j is shear strength along the joint under an effective normal stress, σ_nj. The values of n and r are given in Tables 6, 7 and 8. When friction values are not available from shear tests, the same may be obtained from Table 8 based on intact rock strength. The variation of E_j/E_i with joint factor, J_f, is similar to the theoretical prediction by Walsh and Brace [28] and Hobbs [2] for one-dimensional compression between E_j/E_i versus joint frequency, J_n. But in Eq. (10), J_f involves not only J_n but also inclination of the critical joint and the strength likely to be mobilized along this joint.

Table 6 Values of n for different joint inclination, β° [27, 34]

Table 7 Suggested joint strength parameter, r, for filled-up joints at residual stage [27]

Table 8 Suggested values of r for values of σ_ci [34]

Now from Eqs. (9) and (10), the modulus ratio of the jointed mass with respect to that of the intact rock is given by Eq. (12)

$$ M_{\text{rj}} /M_{\text{ri}} = \exp \left[ { - 0.0035J_{\text{f}} } \right] $$

(12)

Table 9 gives the estimated values of σ_cj and M_rj for different values of J_f varying from 0 to 500 for σ_ci = 100 MPa and M_ri = 500 of intact rock as an example. The M_rj values rapidly decrease with the increase in J_f. This table suggests that the relationship between E_j and σ_cj (i.e., M_rj) cannot be taken as constant or greater than M_ri when the rock mass experiences fracturing and undergoing change to lower quality.

Table 9 Estimation of σ_cj and M_rj from J_f for σ_ci = 100 MPa; assumed M_ri = 500 for intact rock

Classification Based on Strength and Modulus Ratio

Even though the original classification due to Deere and Miller was suggested only for intact rocks, by considering σ_Ci and E_i it was modified to classify rock masses as well [29]. It is a two-lettered classification: First letter suggests the range of σ_C and the second letter the range of M_r. The main advantage of such a classification (Tables 10 and 11) is that it not only takes into account two important engineering properties of the rock mass but also gives an assessment of the failure strain (ε_f) which the rock mass is likely to exhibit in the uniaxial compression, where in the stress–strain response it is nearly linear. That is,

Table 10 Strength classification of intact and jointed rocks

Table 11 Modulus ratio classification of intact and jointed rocks

$$ {\text{Modulus}}\;{\text{Ratio}},\;M_{\text{rj}} = E_{\text{tj}} /\sigma_{\text{cj}} = 1/\varepsilon_{\text{fj}} . $$

(13)

Further, the ratio of the failure strain of the jointed rock to that of the intact rock is given by

$$ \varepsilon_{\text{fi}} /\varepsilon_{\text{fj}} = M_{\text{rj}} /M_{\text{ri}} = \exp \left( { - 3.50 \times 10^{ - 3} J_{\text{f}} } \right). $$

(14)

On the basis of experimental data [27], the following simpler expression was also suggested,

$$ \varepsilon_{\text{fj}} = 50\left( {M_{\text{rj}} } \right)^{ - 0.75} \,{\text{percent}}. $$

(15)

Figure 2 is an extended version of Deere and Miller approach [1] and will cover very low strength-to-very high strength rocks. A modulus ratio of 500 would mean a minimum failure strain of 0.2%, whereas a ratio of 50 corresponds to a minimum failure strain of 2% as per Eq. (13). Very soft rocks and dense/compacted soils would show often failure strains of the order of 2%. Therefore, the modulus ratio of 50 was chosen as the lower limiting value for rocks [29].

Fig. 2

Influence of jointing on modulus ratio [30]

In Fig. 2, the location of the intact specimen is shown at “I” on the σ_ci,j and E_i,j plot. When the experimental data of σ_cj and E _j of the jointed specimens of the same material as that of the intact specimen are plotted, all the points fall along an inclined line originating, say at “I”, cutting across the constant boundaries of modulus ratio. This suggests that as fracturing continues, the locations represented by σ_cj and E_j follow a definite trend [30]. These data are from test specimens, each of which had on an average more than 260 elemental cubes and wedge shape elements. These specimens have undergone either sliding, shearing, splitting or rotational mode of failure.

Unconfined compression tests were also carried out on three weathered rocks, namely quartzite, granite and basalt [31]. These three rocks have gone through different stages of weathering, namely unweathered (i.e., fresh), slightly, moderately, highly and completely weathered. These tests were carried out on five levels of weathering of quartzite and four levels of weathering of both granite and basalt. The values of compressive strength and modulus are presented together for these rocks in Fig. 3. It is interesting to observe that the average line cuts the M_rj = 50 line at about σ_cj = 1 MPa. Therefore, soil–rock boundary is not only when σ_cj = 1 MPa but also when M_rj = 50 and J_f = 300 per meter [27]; that is, for rock mass σ_cj > 1 MPa, M_rj > 50, J_f < 300.

Fig. 3

Influence of weathering on modulus ratio of rocks [31]

Ideally when field tests are conducted, the test block is to be isolated from the parent mass by careful cutting and dressing operations to assess σ_cj and E_j in the unconstrained condition. Such a test block should have a slender ratio more than one, preferably two. Unfortunately, the data from such tests are rarely available. Whenever some data are available, it is projected to indicate the effect of the specimen size rather than the change in the quality of the rock within the test specimen/block. As the size increases, the number of joints, their inclination, even if the strength along some of the joints remains same, would effect the response of the block. If one compares a value reflected by the large sized test blocks to that of the intact specimen, the values particularly σ_cj/σ_ci and E_j/E_i would correspond to higher values of J_f. A more recent example is from Natau et al. [32] whose test results from three sizes of specimens ranging from 80 mm to 620 mm were obtained totally in the unconfined state. The average results of σ_cj and E_j are presented in Table 12. From these results, σ_cj of 620 × 620 × 1200 mm specimen is 0.235 times the value of 80-mm dia. specimen. By extrapolation, the value of compressive strength of NX size, assuming it to represent an intact rock, this ratio works out to be 0.20. The values of σ_ci of the NX size works out to be 50 MPa and E_i= 50 GPa. Similarly, the ratio of E_j of 620-mm specimen to the NX size is 1/20. This ratio suggests an average J_f of 230/meter from strength and modulus considerations as per Eqs. (9, 10). The ratio M_ri of NX size is 1000, and the M_rj of 620-mm-size specimen works out to be 250, suggesting a considerable change in the quality of the rock in the larger size. These data also do confirm that the M_rj values should decrease considerably with the decrease in the quality of the rock and not increase, remain constant or vary marginally. Earlier investigations of Rocha [33] also suggested very low values of E_j/E_i as 1/29 for granite, 1/28 for schist, 1/64 for limestone and 1/108 for quartzite.

Table 12 Size effect on modulus ratio [32]

Confining Pressure Influence

Most of the data of modulus are obtained by conducting tests on limited area in tunnels, in drifts or in boreholes. Even if plate loading tests are conducted on a level surface underground or in open excavation, there is always some degree of lateral confinement. The measured modulus values tend to be higher particularly for weaker rock masses. Such results need to be corrected for lateral confinement to obtain values corresponding to the unconfined condition. When such data are provided, the designer has the freedom to choose or modify the strength and modulus depending upon the in situ stress expected in the field. Using Eq. (16) [34], the influence of confining pressure on E_j can be estimated,

$$ E_{{{\text{j}}0}} /E_{{{\text{j}}3}} = 1{-}\exp \left[ {{-}0.10\sigma_{\text{cj}} /\sigma_{3}^{\prime } } \right], $$

(16)

where the subscripts 0 and 3 refer to σ′₃ = 0 and σ′₃ > 0; σ′₃ is the effective confining stress. For σ_cj = 5 MPa, E_j3 for σ′₃ = 2 MPa confining pressure will be 4 times and for σ′₃ = 1 MPa, it will be 2.3 times of E_j0. This is likely to happen in field plate load tests conducted underground on a limited surface area or when lateral in situ stress not being fully released.

The strength criterion for the jointed rocks when σ′₃ is large compared to its tensile strength, σ_t, is given by

$$ {{\left( {\sigma_{1}^{{\prime }} - \sigma_{3}^{{\prime }} } \right)} \mathord{\left/ {\vphantom {{\left( {\sigma_{1}^{{\prime }} - \sigma_{3}^{{\prime }} } \right)} {\sigma_{3}^{{\prime }} }}} \right. \kern-0pt} {\sigma_{3}^{{\prime }} }} = B_{\text{j}} \left( {\sigma_{\text{cj}} /\sigma_{3}^{{\prime }} } \right)^{{\alpha {\text{j}}}} , $$

(17)

where σ′₁ and σ′₃ are major and minor principal stresses, respectively, σ_cj is the uniaxial compressive strength of jointed rock obtained from Eq. (9), and α_j and B_j are strength parameters of the jointed rock. The values of α_j and B_j are obtained from Eqs. (18, 19),

$$ \alpha_{\text{j}} /\alpha_{\text{i}} = (\sigma_{\text{cj}} /\sigma_{\text{ci}} )^{0.5} \;{\text{and}} $$

(18)

$$ B_{\text{i}} /B_{\text{j}} = 0.13\exp \,[2.04\alpha_{\text{j}} /\alpha_{\text{i}} ], $$

(19)

where α_i and B_i are values of strength parameters obtained from triaxial tests on intact rock specimens for the failure criterion [27, 34]. When Eq. (17) is to be applied for the strength of rock mass along the periphery of excavation, i.e., when $\sigma _{3}^{\prime }~{\underline{\wedge}} 0$, the tensile strength, σ_t, of rock mass has to be considered in the denominator along with σ₃ as per the original expression for the strength of rock [27]. One way to assess σ_t for rock mass would be to consider proportional reduction from the intact rock σ_t, similar to the proportional reduction in compressive strength of intact rock.

Prediction of Rock Mass Responses with Joint Factor

Elasto-plastic Analysis

The power house complex of Nathpa Jhakri hydropower project in North India consists of two major openings, i.e., machine hall 216 × 20 × 49 m³ (length × width × height) with an overburden of 262.5 m and a transformer hall 198 × 18 × 29 m³ which is located downstream of the machine hall. The in situ stress for the rock was determined using hydraulic fracturing technique. The vertical stress was found to be 5.89 MPa with an in situ stress ratio of 0.8035. The constitutive model based on disturbed state concept [35] was used to characterize complete stress–strain behavior of the intact and rock mass. Material parameters for the model were determined for the rock specimens as indicated in Table 13 [36]. The rock mass was discretized into 364 eight nodded elements and 1167 nodes, keeping in view the various stages of excavation.

Table 13 Intact and rock mass properties of Quartz Mica Schist [36]

Strength and modulus of the jointed rock mass, quartz mica, were determined with joint factor J_f [37] to carry out the finite element analysis of the powerhouse cavern by considering strain softening behavior of rock mass. The failure criterion based on J_f [27] was adopted to estimate the strength under different in situ stresses. The analysis was carried out using computer code DSC-SST-2D developed by Desai [38]. Twelve stages of excavation were used in the study.

The predicted values of the displacements by FEM were compared with the observed values at six locations and found to be within the range of measured deformations at five out of six locations (Table 14).

Table 14 Comparison of predicted and observed deformations at the powerhouse cavern boundary, [36]

Equivalent Continuum Modeling (ECM)

Using the strength, modulus and failure strain relations for rock mass with J_f and the corresponding values of intact rock, a few cases were analyzed [39,40,41,42]. The tangent elastic modulus of intact rock was represented by confining stress-dependent hyperbolic relation [43]. A numerical model was developed from an existing finite element code for a nonlinear soil–structure interaction program to account for material nonlinearity of both the intact and jointed rocks.

This model was incorporated in the commercial finite difference code FLAC. A FISH function was written to incorporate joint factor model with Duncan–Chang nonlinear hyperbolic relationships in FLAC. The model has been applied to two large power station caverns, one in Japan and the other in the Himalayas, and to a slope at Kiirunavaara mine in Sweden. For validation purposes, the finite element analysis was applied to predict the response of jointed rocks of sandstones, granite and gypsum plaster and compared with the experimental results. Only sample stress–strain plots for multiple jointed specimens of Agra sandstones are shown in Fig. 4 and for block jointed specimens of gypsum plaster [44] for different confining pressures are shown in Fig. 5 along with the experimental results [40].

Fig. 4

Stress–strain plots for intact and multiple jointed specimens, for confining pressure 5.0 MPa, experimental data after Arora [51]

Fig. 5

Calculated and experimental stress–strain plots for block jointed specimen of gypsum plaster for different confining pressures [41]

Analysis of Shiobara Power House Cavern

The equivalent continuum model was applied for the analysis of a large cavern in jointed rock mass for the Shiobara power station in Japan [45]. The cavern (Fig. 6) measures 161 m length, 28 m width and 51 m height, located at a depth of 200 m below the ground. The three in situ principal stresses were recorded as 5.0, 3.9 and 2.8 MPa. The reported average intact rock compressive strength and elastic modulus were 83.3 MPa and 42.1 GPa, respectively. The rock mass was characterized as rhyolite consisting of platy and columnar joints. The cross section of the cavern along with the location of the MPBXs is shown in Fig. 6. The jointed rock mass surrounding the cavern has been analyzed by the finite element method using the proposed equivalent continuum approach. Equivalent material properties for jointed rock were modeled using the relations with J_f which was taken as 41, 12 and 111 per meter for joint sets I, II and III, respectively. The variation of relative displacement at the end of different excavation steps with progress of cavern excavation with time–displacement measurements along the measurement line agreed well with the observed values. These are presented here only for B110 and B111 in Figs. 7 and 8. The variation of relative displacements from FEM and as measured, [45], at the completion of whole excavation along a measurement line were compared for B117 (Fig. 9) [41]. The parameter r was chosen from Table 8, based on intact rock density.

Fig. 6

Cross section of cavern and location of displacement transducer [45]

Fig. 7

Time history of displacements near the cavern wall along measurement line B110 [41]

Fig. 8

Time history of displacements near the cavern wall along measurement line B111 [41]

Fig. 9

Displacement along measurement line B117 at the completion whole excavation [41]

Analysis of Kiirunavaara Mine, Sweden

The Kiirunavaara mine, which is 4000 m long, with an average width of 90 m, is located 144 km north of the arctic circle in the city of Kiruna in North Sweden. The magnetic iron ore body is relatively strong surrounded by competent quartz porphyry on the hanging wall and syenite porphyry on the footwall. The rock mass has three joint sets. One joint set is oriented roughly parallel to the ore body as the other two strikes obliquely to the ore body. All joints dip fairly steep, 60°–90°. The locations where the first set of cracks were observed in 1985 were mapped by Sjoberg [46].

The value of joint factor (J_f) was obtained as 13. The parameter r was chosen from Table 8. The total size of the model used was 2000 × 1300 m [41]. Sequential mining was simulated in the FLAC by modeling the excavated material with null model and solving after each stage of excavation. Failure was only observed from the concentration of shear strains in the model on the foot wall. The paths of concentrated shear strains represented the failure surface in the model. The failure thus simulated by the numerical model using practical equivalent continuum approach was compared with the failure observations in the field. Shear failure was observed in the footwall of the model while excavating at the mining level of − 586 m, agreeing with the field observations as reported by Sjoberg [46]. Typical failure surface for the mining step of − 586 m is presented in Fig. 10. The intact rock parameters are indicated in Table 15.

Fig. 10

Shear strains for FLAC model for − 586 m mining level [41]

Table 15 Parameters for numerical model of Kiirunavaara mine [40, 46]

Analysis of Nathpa Jhakri Power House Cavern

This cavern was also analyzed by Sitharam and Madhavi Latha [40]. The finite difference grid used for the analysis was of size 210 m × 450 m with 1320 rectangular zones. The excavation steps were simulated in the numerical analysis (FLAC 2D), and the locations of the installation of extensometers were identified for obtaining the displacements for comparison with the measured displacements from instrumentation of the cavern. The variation of displacements with time is also obtained from numerical analysis by solving for equilibrium after each excavation step. Comparison of the observed and predicted deformations along the measurement line at different locations for various excavation levels after the completion of excavation is presented in Table 16. The joint factor was estimated as 22 per meter for the analysis, and the value of r was chosen from Table 8. And other parameters of intact rock were as indicated in Table 13. The ranges of deformations indicate movements of measuring points on the face of the cavern along its length at these elevations. The variation of displacements with time was also obtained from numerical analysis by solving for equilibrium after each stage of excavation step. Figure 11 shows comparison of measured and predicted displacements at two locations behind the face of the power house cavern, with time using joint factor linked relationships [27].

Table 16 Measured and predicted deformations at Nathpa Jhakri cavern [41]

Fig. 11

Comparison of displacements along measurement line for powerhouse cavern [42]

Abutment Stability of Chenab Bridge

Slope stability analysis of the right abutment (359 m high) at Kauri side of Chenab river between Katra and Laole, Jammu and Kashmir, India, was carried our using FLAC of plane strain case for pseudo-static approach with earthquake intensity of 6.5 [47], Sitharam and Maji [48]. With J_f values and intact rock properties, σ_cj, E_j, hyperbolic stress–strain response, c_j and ϕ_j for the rock mass are computed. By applying varying factors of safety to c_j and tan ϕ_j, failure conditions in the slope were determined. The rock parameters adopted were: $\sigma_{\text{ci }} = 115 \,{\text{MPa}},\; E_{\text{i }} = 65\, {\text{GPa}},\;J_{\text{f }} = 320, \;\sigma_{\text{cj }} = 5.38\,{\text{MPa}}, \;E_{\text{j}} = 4.34 \,{\text{GPa}},\; c_{\text{j}} = 1.785\,{\text{MPa}}, \; \phi_{\text{j}} = 23^{ \circ } .$ The static factor of safety achieved was 1.86.

Stress–Strain Response

Arunakumari and Latha [49] introduced user-defined FISH functions to incorporate relations based on J_f into the explicit finite difference code FLAC (version 4.0, ITASCA 1995) to simulate exact joint behavior. Adopting hyperbolic formulation of stress–strain response of jointed rock, estimated elastic initial tangent modulus as per Duncan and Chang [43], they predicted stress–strain, strength envelopes and variation of strength with joint orientation of tested specimens. A very good agreement has been shown.

Joint Factor in ANN Model

By constructing an ANN model, stress–strain response and variation of E_j/E_i with J_f were predicted for jointed rocks, by specifying intact rock properties, σ₃, J_f and axial strain as inputs [50]. Out of number of cases presented by them, only stress–strain curves predicted by equivalent continuum (with J_f) and ANN models are presented in Fig. 12 with experimental results of block jointed gypsum plaster [44]. Figure 13 shows the predictions as per ANN with experimental results of Agra sandstone [51].

Fig. 12

Comparison of stress–strain curves predicted by ECM and ANN with experimental values for block jointed gypsum plaster [50]

Fig. 13

Effect of joint orientation on the stress strain response of Agra sandstone [50]

Comparison of ECM and DCM

More recently, Latha and Garaga [52] have predicted the stress–strain response of jointed specimens of six different rock types with varying joint inclination and joint frequency in triaxial compression adopting equivalent continuum method (ECM) based on joint factor and all the related equations [26, 34] and discrete continuum method (DCM). Both the approaches predict reasonably close responses of jointed specimens and demonstrated that ECM can be adopted without compromising much on the accuracy. They applied ECM to Shimizu tunnel No. 3, which is a part of the new Tomei Express way in Japan. The tunnel length is 1.2 km located at a depth of 83 m below ground level in soft sandstone. Three joint sets exist, namely bedding plane, dipping 28°, random cross joints dipping 58° and near vertical joint dipping 88°. The horizontal stress is 2.03 MPa, and the vertical stress is 1.73 MPa. It was decided to excavate the tunnel in three stages, i.e., pilot tunnel, top heading and bottom benching; the tunnel being 12 m high and 18 m wide. The soft sandstone has $\sigma_{\text{ci }} = 60 \,{\text{MPa}},\;E_{\text{i}} = 3000\, {\text{MPa}},\;c_{\text{i}} = 2\, {\text{MPa}}\; {\text{and}}\; \varPhi_{\text{i}} = 38^{ \circ } .$ The joint factor, $J_{\text{f }}$, worked out for the critical joint dipping at 28° is 111, as per Ramamurthy [26, 34]. The detailed analysis carried out by ECM using FLAC to predict displacements at various locations along the crown shows good agreement with the observed values and those predicted by Vardakos [53] using UDEC, as indicated in Table 17.

Table 17 Comparison of displacements from numerical methods with field measurements [53]

Penetration Rate of TBMs $P_{R}$

Whether it is in $Q_{\text{TBM}}$ [54], RMR or any other rock mass classification linked to $P_{R}$, the modulus of rock has been ignored (Fig. 14). For producing indentation by crushing under the tip of the cutter, compressive and tensile strengths are important. In doing so, whatever deformation/penetration is produced will depend on the modulus response of rock mass. It is therefore very essential that the modulus of rock mass be considered. More precisely, the modulus ratio to account for the combined influence of compressive strength $(\sigma_{\text{cj}} )$ and modulus $(E_{\text{j}} )$ of the rock mass, i.e., $M_{\text{rj}} = E_{\text{j}} /\sigma_{\text{cj}}$. Basically under each cycle of boring by TBM, the various other major factors which control $P_{R}$ are included in the following Eq. (20). This equation is dimensionally correct and predicts $P_{R}$ value per meter of advance of boring as indicated below [55],

$$ P_{R} = \frac{{(T/A) \cdot (\sigma_{\text{ci}} /\sigma_{\text{t}} ) \cdot R \cdot N \cdot ({\text{DRI}}/100) \cdot s}}{{p_{o} \cdot M_{\text{rj}} }} $$

(20)

where T net thrust, T; A area of the cutter head, m²; $\sigma_{\text{ci}}$ compressive strength of intact rock, MPa; $\sigma_{\text{t}}$ tensile strength of intact rock, MPa; R number of rotations of cutterhead, per hour; N number of cutters, per m²; DRI drilling rate index based on compressive strength of intact rock (Fig. 15) NTH [56]; S unit length of drilling, m; p_o mean biaxial stress on the cutting face, T/m² (or taken as density of rock mass times over burden height); $M_{\text{rj}}$ modulus ratio of rock mass, $( = E_{\text{j}} /\sigma_{\text{cj}} )$,

Fig. 14

Scatter of $P_{R}$ with $Q_{\text{TBM}}$ for Maen tunnel

Fig. 15

Compressive strength versus DRI: (1) Granite, Quartzite, Sandstone, Siltstone (coarse to fine grained); (2) Mica schist/Mica gneiss; (3) Phyllite/Shale (extended curve)

In Eq. (20), the influence of seepage pressure is not considered, since most of the seepage pressure is dissipated at the cutting face due to the presence of fractures, joints, etc. The seepage pressure acting through the intact rock will be negligible anyway on the cutting face. The rock parameters are to be obtained under saturated condition, if seepage exists. If seepage pressure exists, effective trust should be considered

The ratio $(\sigma_{\text{ci}} /\sigma_{\text{t}} )$ takes care of inherent anisotropy in the intact rock and also its brittleness. When the gouge material thickness is less than 5 mm, the blocks formed in the rock mass may remain tight/interlocked and may not get dislodged during boring operation. But when the gouge material thickness is more than 5 mm, rock blocks get dislodged and may damage the cutters. To take into consideration the thickness of gouge in the estimation of $J_{\text{f}} ,$ equivalent number of joints are estimated by dividing the thickness of gouge (in mm) by 5 mm, which is the minimum thickness of gouge to be effective [27].

Case Study for Penetration Rate

Excellent data were collected by Sapigni et al. [57] from NW Alps. These data are applied to verify Eq. (20). What all the data are required for this presentation is reported by Sapigni et al. for metabasite in Maen tunnel and for micaschist and metadiorite in Pieve tunnel for various values of RMR. The scatter of $P_{R}$ values with Q for these rock types are shown in Fig. 15. The RMR values given have been converted to $J_{\text{f}} ,$ joint factor. The values of compressive strength, tensile strength and modulus values for the three rocks are given in Table 18. The basic tunnel equipment data of Maen tunnel and that of Pieve tunnel are given in Table 19 for use in Eq. (20). Table 20 gives minimum, average, maximum values of $M_{\text{ri}}$ as per Table 18 and also values of $M_{\text{rj}} /M_{\text{ri}}$ for different values of RMR and $J_{\text{f}}$. Tables 21, 22 and 23 present actual range of $P_{R}$ versus $J_{\text{f}}$, RMR and the values of $P_{R}$ estimated from Eq. (20) for minimum, average and maximum values of $M_{\text{rj}}$. A comparison of the calculated and field measured mean $P_{R}$ values in these three tables for 16 rock types clearly suggests a good agreement. By considering lower $M_{\text{ri}}$ values, the $P_{R}$ values nearly match with the maximum values of $P_{R}$ from field. Adoption of maximum values of $M_{\text{rj}}$ will suggest lower values in the range. With the adoption of average values of $M_{\text{ri}}$ in Eq. (20), the $P_{R}$ values estimated would suggest mean $P_{R}$ values from the field. In Table 19 for Maen tunnel, the rotation of cutterhead has been at two rates, namely 660 rph and 330 rph. By considering rph of 330 particularly for $J_{\text{f}}$ of 170 and 225, the $P_{R}$ values will be halved and will be within the actual range in the field.

Table 18 Properties of intact rocks, Sapigni et al. [57]

Table 19 Tunnel Equipment Data (Sapigni et al. [57])

Table 20 Values of M_rj/M_ri for J_f (or RMR)

Table 21 Mean tunnel, metabasite, rock mass density—2.6 T/m³, average overburden—400 m, cutterhead rotation—660 rph

Table 22 Pieve tunnel, micaschist, rock mass density—2.6 T/m³, average overburden—500 m, cutterhead rotation—678 rph

Table 23 Pieve tunnel, metadiorite, rock mass density—2.6 T/m³, average overburden—500 m, cutterhead rotation—678 rph

It has been reported by many investigators that particularly for $J_{\text{f}} > 200$ decrease in $P_{R}$ is generally observed. This is mainly due to the dislodging of rock blocks hindering $P_{R}$ values. Such decrease in $P_{R}$ is indicated from the data of Sapigni et al. [57]. In such situations, the operators usually reduce the rotations of TBM.

The special advantage of adopting Eq. (20) for predicting $P_{R}$ is that all the input data are factual and from test conducted on the rocks as per approved practice. It is dimensionally correct compared to other prevailing expressions. The $P_{R}$ may be calculated per meter of boring in a specified length having similar formation. Assessment of $P_{R}$ per meter length of tunnel is specified because the $J_{\text{f}}$ value is estimated per meter length. On the basis of this, one could estimate average $P_{R}$ in each zone and then an overall estimation of the $P_{R}$ or for the entire length of the tunnel would result. Since an excellent site investigation of a tunnel alignment is essential for its successful execution with TBM, Eq. (20) will certainly be very handy in predicting $P_{R} .$

Discussion

Equation (20) is simple and dimensionally in order unlike all other expressions in vogue to predict $P_{R}$. It takes into consideration the basic parameters of TBM, intact rock, rock mass, in situ stress and drilling rate index which is linked to compressive strength of intact rock. The prediction of $P_{R}$ has been made for 16 rock types from Mean and Pieve tunnels, and a very good agreement is observed with the field data. The thrust of TBMs as given for these tunnels has been adopted in the estimation of $P_{R}$; it is the maximum thrust. The net values of thrust may be some what less by 10–15%; the $P_{R}$ values will not be drastically altered. At the most, a correction factor of 1.2–1.5 may have to be applied in Eq. (20) to obtain more realistic values of $P_{R}$ by considering net thrust. Rest of the data adopted are either measured or taken from well-established relationships. A reduction of 10%, 25% and 40% in the calculated $P_{R}$ may have to be made for values of $J_{\text{f}}$ values of 250, 300 and 350, respectively, to account for the decrease in $P_{R}$ for $J_{\text{f}}$ > 200 due to dislodging of rock blocks and slow down of TBM rotations.

Conclusions

A critical examination of the most commonly adopted rock mass classifications, namely RMR, Q and GSI, has revealed that the compressive strength and modulus values suggested need some definite modifications based on the modulus ratio criterion, which defines the quality of rock mass. In practice, the modulus ratios of rock masses have been found to be much higher than those of intact rocks. Predicted deformations did not agree with the field measured values. Application of joint factor, J_f, to solve some field problems and prediction of the response of laboratory tests seems to be encouraging. The relationships based on J_f for rock masses appear to be more realistic since these are based on experimental verifications. A unified rock mass classification based on modulus ratio concept, applicable to both intact and mass of rocks, would give better assessment of engineering responses. Consideration of modulus ratio in predicting the penetration rate of TBMs seems to be more realistic, simple and easy to apply.